Representation Theory of Finite Monoids

Steinberg, Benjamin

doi:10.1007/978-3-319-43932-7

Cited by 122 publications

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“…The book [38] serves as a basic reference for those aspects of the representation theory of monoids that we shall need. If M is a finite monoid, then CM denotes the monoid algebra of M .…”

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

“…The character of a simple module is called an irreducible character. The irreducible characters of a monoid form a linearly independent set of mappings [38,Theorem 7.7].…”

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

“…The fundamental theorem of Clifford-Munn-Ponizovskii theory says the following. See [23] or [38,Theorem 5.5] for details. McAlister [29] gave a general method to compute the composition factors of a module from its character by inverting the character table of the monoid.…”

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

“…If we identify P with the element m∈M P (m)m ∈ CM, then the transition matrix of the random walk is the transpose of the matrix of the operator P acting on the vector space CΩ with respect to the basis Ω. See [38,Chapter 14] or [8,16,17] for details. Note that the tranpose arises here because we are using row stochastic matrices for the transition matrix but left actions for the random walk.…”

Section: 3mentioning

confidence: 99%

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Random walks on rings and modules

Ayyer¹,

Steinberg²

2020

Algebraic Combinatorics

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We consider two natural models of random walks on a module V over a finite commutative ring R driven simultaneously by addition of random elements in V , and multiplication by random elements in R. In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements a ∈ R, b ∈ V are sampled independently, and the current state x is taken to ax + b. For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on V under a suitable hypothesis on the measure on V (the measure on R can be arbitrary).

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“…The book [38] serves as a basic reference for those aspects of the representation theory of monoids that we shall need. If M is a finite monoid, then CM denotes the monoid algebra of M .…”

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

“…The character of a simple module is called an irreducible character. The irreducible characters of a monoid form a linearly independent set of mappings [38,Theorem 7.7].…”

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

Section: Preliminaries On Group and Monoid Representation Theorymentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

See 2 more Smart Citations

Random walks on rings and modules

Ayyer¹,

Steinberg²

2020

Algebraic Combinatorics

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“…• There is a labelled edge (U ,V ) a − → (U ′ ,V ′ ) for each pair of vertices (U ,V ) and (U ′ ,V ′ ), and each matrix a ∈ A such that ker(a) = U and im(a) = V ′ . We note in passing that K(A) can be seen as an edge-induced subgraph of the Karoubi envelope [41] of the semigroup M n (C).…”

Section: A Generating Graphmentioning

confidence: 99%

Polynomial Invariants for Affine Programs

Hrushovski

Ouaknine

Pouly

et al. 2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

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9 The Representation Theory of Inverse Monoids

Steinberg

2016

Representation Theory of Finite Monoids

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This paper proposes the conjecture that if the Cartan matrix of the complex algebra of a finite monoid is nonsingular, then the Cartan matrix of the algebra of that monoid over any field is nonsingular. The conjecture is proved for von Neumann regular monoids and for monoids whose left (or right) stabilizers are aperiodic. Both these classes contain the class of finite groups, for which the conjecture holds by classical results of Brauer. The basics of modular representation theory for finite monoids are developed in order to elaborate on this conjecture and to prove some of the partial results.

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Representation Theory of Finite Monoids

Cited by 122 publications

References 0 publications

Random walks on rings and modules

Random walks on rings and modules

Polynomial Invariants for Affine Programs

9 The Representation Theory of Inverse Monoids

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